Question

1.33 Electrons in molecules are described by wavefunctions that extend over more than one atom. Consider an electron that is described by a wavefunction that extends over two adjacent carbon atoms. The electron can move freely between the two atoms. The internuclear C-C distance is 139 pm. (a) Using the one-dimensional particle-in-the-box model, calculate the energy required to promote an electron from the n=1 to the n = 2 level, assuming that the length of the box is the same as the distance between the two carbon atoms. (b) To what wavelength of radiation does this correspond? (c) If each atom in a linear chain of 10 carbon atoms contributes one electron, what is the minimum number of wavefunctions required to account for all the electrons? (d) Repeat the calculation in (a) for a linear chain of 10 carbon atoms with the same internuclear C—C distance (139 pm) but in which the transition takes place from the uppermost filled level to the one above (e) To what wavelength of radiation does the transition in part (d) correspond? (f) A certain compound with a long chain of carbon atoms is found to require light of 696 nm to promote an electron from the n = 6 to the n = 7 level. How long is the chain of carbon atoms in this molecule?

Answer 1

for a partide in a 10-bom; the energy lo expression is is ! Lohn 1: Mass Icelyt 2 3 h (a) of tq.f.: Ah- 9.35 x 1018 Joule $24.11A10-31x(13913 10 24 - he m. = = : -613210-34 x 3.0 x 103 4.35*10-13 -126 x 10-5 m . 2126 *10-4 m 21.26 nm. since each carbon contribulus one electron and each at least one electron is required to construct a wave function; at least 10 ware - functions are required (d) from the above diagram; distance of bon = distance between 1 y 10 - internuclear distance x (107) - 4 x 134 x 10-12 m = 1.251x1004 m The highest filled state has a quantum number of 7:10 i E. ( 1h! llojsh? smis &m² 8m 12 8x9.11610-31 x = $.08 x 1019 J. (1.251x109 21 h 21x16677) 410 Omla m 3 5.05 ev, 6.627x10-34 * 3.0X10 (e) a = he/ B.05x1049 8.09 X 10-19 = 2.734610-8 m = 27.34 nm

- MRT - MRT (f) let the bom-length bel bLZ 49h 8 m 2 or; he of; o L he og hennes 849-11 110-31, 36110 ot; L = 13* 6.629 x 696% 18- 8x9-11X3.0 X10-13 It; 1: 13 X 6,629 %696 x 10-20 Pa Dr; 1 = 2.74x10-13 egg mi? or; 163x10-9.on liven; internuclear distance: 131110-12 na det n be the chain length... Thus; (n-1) % 13 9 x 1012 : 1.65 x 10 9 !, -l : 119 12 or in = 131 ) chain length 14 13.